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arxiv: 2604.12124 · v1 · submitted 2026-04-13 · 📊 stat.ME

Causal inference for spatiotemporal point processes in the presence of outcome spillover and carryover

Conor Kresin , Duncan A. Clark , Louis Davis , Martin Hazelton This is my paper

Pith reviewed 2026-05-10 14:53 UTC · model grok-4.3

classification 📊 stat.ME
keywords causal inferencespatiotemporal point processesoutcome spillovercarryover effectsstochastic EMPoisson processesHawkes processesseismic activity
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The pith

Expected event counts in spatiotemporal point processes are identified under treatment allocations with spillover and carryover.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a causal framework for outcomes that are continuous space-time point processes, such as seismic events or disease cases, where cell-level interventions can spill over to neighboring regions and carry over in time. It represents the observed post-treatment events as an unlabelled superposition of two latent processes, one reflecting the control regime and one the treated regime. On the support of the observed treatment design, expected counts in any spacetime region are identified from standard assumptions of consistency, exchangeability, and positivity. Off-support contrasts rely on a regime-stable structural model of the point process. Estimation proceeds via likelihood with stochastic EM and comes with nonasymptotic contraction bounds for a blockwise hard-EM surrogate, yielding plug-in guarantees for causal functionals.

Core claim

The observed post-treatment process is represented as an unlabelled superposition of latent control and treatment components. On the observed design support, expected post-treatment event counts in any spacetime region under a given treatment allocation are identified under consistency, exchangeability, and positivity; off-support contrasts are identified relative to a regime-stable structural point-process model. Estimation is likelihood-based and implemented with stochastic EM, with nonasymptotic contraction guarantees for a blockwise hard-EM surrogate. The framework covers history-dependent processes including Poisson and Hawkes models.

What carries the argument

Unlabelled superposition of latent control and treatment point-process components, which decomposes the observed intensity to identify potential-outcome expectations indexed by full treatment allocations.

If this is right

  • Plug-in estimators for cell-level and global causal functionals inherit nonasymptotic guarantees from the contraction of the blockwise hard-EM surrogate.
  • The same identification and estimation apply directly to history-dependent processes such as Poisson and Hawkes models.
  • Additional array conditions suffice for unnormalised growing-window contrasts.
  • The approach yields identified causal effects of wastewater injection on seismic activity in the Oklahoma application.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar superposition representations could be tested in other marked point-process settings, such as financial transaction streams, to separate baseline from intervention-driven events.
  • Sensitivity checks that perturb the regime-stability assumption in simulation would quantify how much off-support identification depends on that modelling choice.
  • The framework's handling of spillover suggests direct extensions to interference in network point processes where nodes influence one another's event rates.

Load-bearing premise

The observed events arise as an unlabelled mixture of control and treated latent processes whose off-support contrasts depend on a regime-stable model that is not directly testable from the data.

What would settle it

A dataset in which control and treatment events are distinguishable by mark or timing, yet the estimated causal contrasts diverge from those obtained by treating the events as an unlabelled superposition, would show the identification strategy fails.

Figures

Figures reproduced from arXiv: 2604.12124 by Conor Kresin, Duncan A. Clark, Louis Davis, Martin Hazelton.

Figure 1
Figure 1. Figure 1: Discrete-time motivational DAG with temporal dependence and cross-cell outcome spillover. The cross-cell arrows show why later outcomes depend on the full treatment allocation. The boxed lagged outcomes Y 2 1 and Y 2 2 are the post-treatment variables one might be tempted to condition on; doing so blocks some directed spillover paths to later outcomes but can also open noncausal paths through collider stru… view at source ↗
Figure 2
Figure 2. Figure 2: Estimated observed-regime vs no-directive DAITE under oracle and naive labelling and the full SEM procedure. The dashed line marks the true value. 0 10 20 −250 0 250 500 750 bootstrap replicate total saved / 100 day Count Model Naive Bivaraite ETAS with KDE SEM Bivariate ETAS with KDE (a) Bootstrap replicate distribution of all-or-nothing total saved for naive vs. SEM. The dashed vertical line marks zero s… view at source ↗
Figure 3
Figure 3. Figure 3: Oklahoma results. We restrict to earthquakes of magnitude 2.5 or greater and consider bivariate epidemic-type aftershock sequence (ETAS) models (Ogata, 1988) for latent control and treated components with a shared triggering kernel and KDE-driven inhomoge￾neous background (Nandan et al., 2017). We use the first half of the pre-treatment data to estimate the background rate and proceed with our SEM approach… view at source ↗
Figure 4
Figure 4. Figure 4: One realisation of control and treatment process points superposed the full SEM estimation method. Due to the presence of pre-treatment data, the control parameters are used at their true values in the calculation of the DAITE. The naive labelling underestimates the effect of treating all regions because spillover from high-productivity control regions inflates the estimated K for the treated process. In c… view at source ↗
Figure 5
Figure 5. Figure 5: Estimated all-or-nothing DAITE under oracle and naive labelling and the full SEM procedure. The dashed line marks the true value [PITH_FULL_IMAGE:figures/full_fig_p028_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Oklahoma and the AOI marked with dashed lines, counties are considered treated if their centroid is within the AOI Let z ∈ {0, 1} index latent process. The conditional intensity for component k ∈ {0, 1} takes the form λk(t, x, y) = µk |Sk| Wk(x, y)+ X ℓ∈{0,1} X j:tj<t rj=ℓ Akℓ e αm,kℓ(mj−m0) g(t−tj ) f(x−xj , y−yj | mj ), with Omori-Utsu g(∆t) = p−1 c (1 + ∆t/c) −p on ∆t > 0 and power-law spatial factor f … view at source ↗
Figure 7
Figure 7. Figure 7: Pre and post-treatment point pattern with colour denoting location inferred process density values are normalized cell-wise so that background mass matches each cell’s area. S3.5. Stochastic EM configuration and Diagnostics The SEM KDE fit requires tuning and uses the hyper parameters described in [PITH_FULL_IMAGE:figures/full_fig_p031_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: SEM Diagnostics for Oklahoma KDE SEM fit [PITH_FULL_IMAGE:figures/full_fig_p032_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Dependency structure of the analysis. The diagram flows from top (assumptions) to bottom (theorems). Dashed boxes indicate logical groupings [PITH_FULL_IMAGE:figures/full_fig_p049_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Predictable blockwise penalized hard-EM operator used in the analysis. The practical SEM implementation may include additional offline smoothing passes, but the theoretical argument only requires the time-respecting operator displayed here. current block), set r (m+1) i ∈ arg max k∈{0,1} n log λbr (m+1) k (γi ; θ (m) ) − αKwin 1 n γi ∈ S + θ (m) (b) ∪ S − θ (m) (b), k = minority(γi ; θ (m) , b) o o. (17) … view at source ↗
read the original abstract

We develop a framework for causal inference with continuous spatiotemporal point-process outcomes under cell-level interventions and outcome spillover. Potential outcomes are indexed by full treatment allocations, and the observed post-treatment process is represented as an unlabelled superposition of latent control and treatment components. On the observed design support, expected post-treatment event counts in any spacetime region under a given treatment allocation are identified under consistency, exchangeability, and positivity; off-support contrasts are identified relative to a regime-stable structural point-process model. Estimation is likelihood-based and implemented with stochastic EM. To understand when this is feasible, we analyse a predictable blockwise hard-EM surrogate and show nonasymptotic contraction of estimation error to a statistical floor governed by locally ambiguous regions. This yields plug-in guarantees for cell-level and global causal functionals, and clarifies the additional array conditions needed for unnormalised growing-window contrasts. The framework covers history dependent spatiotemporal point processes including Poisson and Hawkes models, with applications to settings such as epidemiology, seismology, and finance. We provide an application assessing the causal effect of injecting wastewater into the ground on seismic activity in Oklahoma.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a causal inference framework for continuous spatiotemporal point processes under cell-level interventions with outcome spillover and carryover. Potential outcomes are indexed by full treatment allocations, and the observed post-treatment process is modeled as an unlabelled superposition of latent control and treatment components. On the observed design support, expected post-treatment event counts in any spacetime region are identified from consistency, exchangeability, and positivity alone. Off-support contrasts are identified relative to a regime-stable structural point-process model. Estimation is likelihood-based via stochastic EM, with nonasymptotic contraction analysis for a predictable blockwise hard-EM surrogate that yields plug-in guarantees for cell-level and global functionals. The framework applies to Poisson and Hawkes processes and is illustrated with an application to wastewater injection and seismic activity in Oklahoma.

Significance. If the regime-stable structural model is appropriate, the work provides a coherent extension of causal identification and estimation to dependent point-process outcomes with spillover, addressing a gap in settings such as epidemiology and seismology. The nonasymptotic contraction result for the EM surrogate and the explicit array conditions for growing-window contrasts are genuine strengths that go beyond standard asymptotic arguments and support reproducible plug-in inference.

major comments (2)
  1. [Identification and off-support contrasts] The identification of off-support contrasts (abstract and identification section) rests on the regime-stable structural point-process model whose kernel and baseline parameters are assumed invariant across treatment allocations. Because the observed process is an unlabelled superposition, any decomposition into latent components is unique only up to the parametric restrictions of this model; the manuscript does not supply a concrete test or sensitivity analysis for local misspecification of the stability assumption, which directly undermines the plug-in guarantees for any contrast that extrapolates beyond observed design support.
  2. [Estimation and theoretical analysis] § on nonasymptotic analysis: the contraction result for the predictable blockwise hard-EM surrogate is stated, yet the precise conditions required for unnormalised growing-window contrasts (including the array conditions mentioned in the abstract) are not fully verifiable from the provided description. These conditions are load-bearing for the claimed plug-in guarantees on global functionals and must be stated explicitly with the relevant measurability and boundedness assumptions.
minor comments (2)
  1. [Model setup] Notation for the latent control and treatment components would benefit from an explicit diagram or table clarifying the superposition mapping and the regime-stability restrictions.
  2. [Application] The application section would be strengthened by reporting the specific parametric form chosen for the regime-stable Hawkes kernel and any diagnostic checks performed on the stability assumption.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive suggestions, which have prompted us to strengthen the presentation of our identification and estimation results. We respond to each major comment below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [Identification and off-support contrasts] The identification of off-support contrasts (abstract and identification section) rests on the regime-stable structural point-process model whose kernel and baseline parameters are assumed invariant across treatment allocations. Because the observed process is an unlabelled superposition, any decomposition into latent components is unique only up to the parametric restrictions of this model; the manuscript does not supply a concrete test or sensitivity analysis for local misspecification of the stability assumption, which directly undermines the plug-in guarantees for any contrast that extrapolates beyond observed design support.

    Authors: We agree that the regime-stable structural model is a substantive modeling assumption required to identify off-support contrasts, since the unlabelled superposition alone does not yield unique latent-component decompositions. The manuscript presents identification results conditional on this invariance of kernel and baseline parameters. While we do not claim the assumption is testable from the observed data without additional structure, we acknowledge that readers would benefit from guidance on its plausibility. In the revision we will add a dedicated sensitivity subsection that perturbs the kernel parameters within ranges consistent with the observed design support and reports the resulting variation in the extrapolated contrasts. This will not alter the formal identification statements but will make the practical scope of the plug-in guarantees more transparent. revision: yes

  2. Referee: [Estimation and theoretical analysis] § on nonasymptotic analysis: the contraction result for the predictable blockwise hard-EM surrogate is stated, yet the precise conditions required for unnormalised growing-window contrasts (including the array conditions mentioned in the abstract) are not fully verifiable from the provided description. These conditions are load-bearing for the claimed plug-in guarantees on global functionals and must be stated explicitly with the relevant measurability and boundedness assumptions.

    Authors: We appreciate the referee’s observation that the array conditions supporting the nonasymptotic contraction for unnormalised growing-window contrasts are referenced in the abstract but not stated with full precision in the main text. The analysis relies on standard measurability of the filtration, uniform boundedness of the intensity functions on compact sets, and specific moment-array conditions that ensure the contraction mapping applies uniformly. In the revised manuscript we will restate the relevant theorem with an explicit list of these assumptions (including the precise array conditions) and expand the proof sketch to verify each one. This change will make the plug-in guarantees for global functionals directly verifiable without altering the underlying contraction result. revision: yes

Circularity Check

0 steps flagged

No significant circularity: identification rests on external assumptions and estimation uses standard latent-variable methods with independent guarantees

full rationale

The derivation chain begins with standard causal assumptions (consistency, exchangeability, positivity) to identify on-support expectations for post-treatment event counts under a given allocation; these are not derived from or equivalent to the observed superposition by construction. Off-support contrasts are identified only relative to an additional regime-stable structural point-process model (covering Poisson and Hawkes), which is posited as an untestable modeling assumption rather than recovered from data or fitted parameters. Likelihood-based estimation via stochastic EM, together with the nonasymptotic contraction analysis of the blockwise hard-EM surrogate, yields plug-in guarantees whose error floor is governed by locally ambiguous regions; this analysis does not reduce the target causal functionals to the fitted quantities tautologically. No self-citation load-bearing steps, uniqueness theorems imported from prior author work, or ansatz smuggling appear in the identification or estimation claims. The framework is therefore self-contained against external benchmarks once the stated assumptions are granted.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on three standard causal assumptions plus a regime-stable structural model and the latent superposition representation. No free parameters are explicitly fitted in the abstract; the structural model may introduce domain-specific assumptions.

axioms (2)
  • domain assumption Consistency, exchangeability, and positivity for identification on observed design support
    Invoked for identification of expected event counts under observed treatment allocations.
  • ad hoc to paper Regime-stable structural point-process model for off-support contrasts
    Required to identify contrasts for treatment allocations outside the observed design support.
invented entities (1)
  • Latent control and treatment components no independent evidence
    purpose: Represent the observed post-treatment process as an unlabelled superposition
    New modeling device that allows separation of potential outcomes in the point-process setting

pith-pipeline@v0.9.0 · 5497 in / 1519 out tokens · 53815 ms · 2026-05-10T14:53:02.843901+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

3 extracted references · 3 canonical work pages

  1. [1]

    AOI region

    andλ 1(· |θ ′ 1), compute predicted post- treatment countsC j,0 andC j,1 (via compensator integrals or fast simulation), and define D+ j = Cobs j −C j,0 +, D − j = Cj,0 −C obs j +. LargeD + j flags excess mass the control model cannot explain and suggests flipping control labels to treatment inI j;D − j suggests the reverse. A single proposal is generated...

    work page 2025

  2. [2]

    Throughout the analysis, we index complete-data likelihoods and intensities by a labelling r= (r i)i≥1 ∈ {0,1} N

    For any predictable setS⊂D, define the component compensated martingales M ⋆ k (S) :=N k(S)− Z S λ⋆ k(τ)dτ, k∈ {0,1}, and the superposed compensated martingale M ⋆(S) :=N(S)− Z S λ⋆(τ)dτ=M ⋆ 0 (S) +M ⋆ 1 (S). Throughout the analysis, we index complete-data likelihoods and intensities by a labelling r= (r i)i≥1 ∈ {0,1} N. Only the firstN(D) entries affect ...

    work page 1975

  3. [3]

    log bλr′ ri(γi) bλr′ (γi) −log bλr ri(γi) bλr(γi) # +

    For the remainder of this proof, we writeD:=D (m) for brevity. Forv∈[0,1] also define the interpolated intensity bλr,v(τ) := bλr(τ) +v∆ bλr(τ). Using the decomposition X k∈{0,1} Z D logbλr k(τ)dN r k(τ) = Z D logbλr(τ)dN(τ) + X k∈{0,1} Z D log bλr k(τ) bλr(τ) dN r k(τ), we may write the objective (restricted toD) as a sum of a total-intensity log- likelih...

    work page 2007